Category Archives: Algebraic Geometry

This week I’m down in Canterbury for a conference focussing on the positive Grassmannian. “What’s that?”, I hear you ask. Roughly speaking, it’s a mysterious geometrical object that seems to crop up all over mathematical physics, from scattering amplitudes to solitons, not to mention quantum groups. More formally we define

We can view this as the space of matrices modulo a action, which has homogeneous “Plücker” coordinates given by the minors. Of course, these are not coordinates in the true sense, for they are overcomplete. In particular there exist quadratic Plücker relations between the minors. In principle then, you only need a subset of the homogeneous coordinates to cover the whole Grassmannian.

To get to the positive Grassmannian is easy, you simply enforce that every minor is positive. Of course, you only need to check this for some subset of the Plücker coordinates, but it’s tricky to determine which ones. In the first talk of the day Lauren Williams showed how you can elegantly extract this information from paths on a graph!

In fact, this graph encodes much more information than that. In particular, it turns out that the positive Grassmannian naturally decomposes into cells (i.e. things homeomorphic to a closed ball). The graph can be used to exactly determine this cell decomposition.

And that’s not all! The same structure crops up in the study of quantum groups. Very loosely, these are algebraic structures that result from introducing non-commutativity in a controlled way. More formally, if you want to quantise a given integrable system, you’ll typically want to promote the coordinate ring of a Poisson-Lie group to a non-commutative algebra. This is exactly the sort of problem that Drinfeld et al. started studying 30 years ago, and the field is very much active today.

The link with the positive Grassmannian comes from defining a quantity called the quantum Grassmannian. The first step is to invoke a quantum plane, that is a -dimensional algebra generated by with the relation that for some parameter different from . The matrices that linearly transform this plane are then constrained in their entries for consistency. There’s a natural way to build these up into higher dimensional quantum matrices. The quantum Grassmannian is constructed exactly as above, but with these new-fangled quantum matrices!

The theorem goes that the torus action invariant irreducible varieties in the quantum Grassmannian exactly correspond to the cells of the positive Grassmannian. The proof is fairly involved, but the ideas are rather elegant. I think you’ll agree that the final result is mysterious and intriguing!

And we’re not done there. As I’ve mentioned before, positive Grassmannia and their generalizations turn out to compute scattering amplitudes. Alright, at present this only works for planar super-Yang-Mills. Stop press! Maybe it works for non-planar theories as well. In any case, it’s further evidence that Grassmannia are the future.

From a historical point of view, it’s not surprising that Grassmannia are cropping up right now. In fact, you can chronicle revolutions in theoretical physics according to changes in the variables we use. The calculus revolution of Newton and Leibniz is arguably about understanding properly the limiting behaviour of real numbers. With quantum mechanics came the entry of complex numbers into the game. By the 1970s it had become clear that projectivity was important, and twistor theory was born. And the natural step beyond projective space is the Grassmannian. Viva la revolución!

I’ll start this post by tying up some loose ends from last time. Before we get going there’s no better recommendation for uplifting listening than this marvellous recording. Hopefully it’ll help motivate and inspire you (and I) as we journey deeper into the weird and wonderful world of algebra and geometry.

I promised a proof that for algebraically closed fields every Zariski open set is dense in the Zariski topology. Quite a mouthful at this stage of a post, I admit. Basically what I’m showing is that Zariski open sets are really damn big, only in a mathematically precise way. But what of this ‘algebraically closed’ nonsense? Time for a definition.

Definition 3.1 A field is algebraically closed if every nonconstant polynomial in has a root in .

Let’s look at a few examples. Certainly isn’t algebraically closed. Indeed the polynomial has no root in . By contrast is algebraically closed, by virtue of the Fundamental Theorem of Algebra. Clearly no finite field is algebraically closed. Indeed suppose then has no root in . We’ll take a short detour to exhibit another large class of algebraically closed fields.

Definition 3.2 Let be fields with . We say that is a field extension of and write for this situation. If every element of is the root of a polynomial in we call an algebraic extension. Finally we say that the algebraic closure of is the algebraic extension of which is itself algebraically closed.

(For those with a more technical background, recall that the algebraic closure is unique up to -isomorphisms, provided one is willing to apply Zorn’s Lemma).

The idea of algebraic closure gives us a pleasant way to construct algebraically closed fields. However it gives us little intuition about what these fields ‘look like’. An illustrative example is provided by the algebraic closure of the finite field of order for prime. We’ll write for this field, as is common practice. It’s not too hard to prove the following

Now we’ve got a little bit of an idea what algebraically closed fields might look like! In particular we’ve constructed such fields with characteristic for all . From now on we shall boldly assume that for our purposes

every field is algebraically closed

I imagine that you may have an immediate objection. After all, I’ve been recommending that you use to gain an intuition about . But we’ve just seen that is not algebraically closed. Seems like we have an issue.

At this point I have to wave my hands a bit. Since is a subset of we can recover many (all?) of the geometrical properties we want to study in by examining them in and projecting appropriately. Moreover since can be identified with in the Euclidean topology, our knowledge of is still a useful intuitive guide.

However we should be aware that when we are examining affine plane curves with they are in some sense dimensional objects – subsets of . If you can imagine dimensional space then you are a better person than I! That’s not to say that these basic varieties are completely intractable though. By looking at projections in and we can gain a pretty complete geometric insight. And this will soon be complemented by our burgeoning algebraic understanding.

Now that I’ve finished rambling, here’s the promised proof!

Lemma 3.4 Every nonempty Zariski open subset of is dense.

Proof Recall that is a principal ideal domain. Thus any ideal may be written . But algebraically closed so splits into linear factors. In other words . Hence the nontrivial Zariski closed subsets of are finite, so certainly the Zariski open subsets of are dense.

I believe that the general case is true for the complement of an irreducible variety, a concept which will be introduced next. However I haven’t been able to find a proof, so have asked here.

How do varieties split apart? This is a perfectly natural question. Indeed many objects, both in mathematics and the everyday world, are made of some fundamental building block. Understanding this ‘irreducible atom’ gives us an insight into the properties of the object itself. We’ll thus need a notion for what constitutes an ‘irreducible’ or ‘atomic’ variety.

Definition 3.5 An affine variety is called reducible if one can write with proper subsets of . If is not reducible, we call it irreducible.

This seems like a good and intuitive way of defining irreducibility. But we don’t yet know that every variety can be constructed from irreducible building blocks. We’ll use the next few minutes to pursue such a theorem.

As an aside, I’d better tell you about some notational confusion that sometimes creeps in. Some authors use the term algebraic set for my term affine variety. Such books will often use the term affine variety to mean irreducible algebraic set. For the time being I’ll stick to my guns, and use the word irreducible when it’s necessary!

Before we go theorem hunting, let’s get an idea about what irreducible varieties look like by examining some examples. The ‘preschool’ example is that is reducible, for indeed . This is neither very interesting nor really very informative, however.

A better example is the fact that is irreducible. To see this, recall that earlier we found that the only proper subvarieties of are finite. But is algebraically closed, so infinite. Hence we cannot write as the union of two proper subvarities!

What about the obvious generalization of this to ? Turns out that it is indeed true, as we might expect. For the sake of formality I’ll write it up as a lemma.

Lemma 3.6 is irreducible

Proof Suppose we could write . By Lemma 2.5 we know that . But so again by Lemma 2.5. Conversely if then either or , so . This shows that .

Now immediately tells us . Suppose that is nonzero. We’ll prove that is the zero polynomial by induction on . Then so not irreducible, as required.

We first note that since algebraically closed infinite. For suppose . Then are each zero at finite sets of points. Thus since infinite, is not the zero polynomial, a contradiction.

Now let . Consider nonzero polynomials in . Fix . Then polynomials in . For some , nonzero as polynomials in . By the induction hypothesis . This completes the induction.

I’ll quickly demonstrate that is quite strange, when considered as a topological space with the Zariski topology! Indeed let and be two nonempty open subsets. Then . Otherwise would be proper closed subsets (affine subvarieties) which covered , violating irreducibility. This is very much not what happens in the Euclidean topology! Similarly we now have a rigorous proof that an open subset of is dense. Otherwise and would be proper subvarieties covering .

It’s all very well looking for direct examples of irreducible varieties, but in doing so we’ve forgotten about algebra! In fact algebra gives us a big helping hand, as the following theorem shows. For completeness we first recall the definition of a prime ideal.

Definition 3.7 is a prime ideal in iff whenever we have or . Equivalently is prime iff is an integral domain.

Proof [““] Suppose not prime. Then with but . Let and . Further define . Then so proper subsets of . On the other hand . Indeed if then so or so .

[““] Suppose is reducible, that is proper subvarieties of with . Since a proper subvariety of there must exist some element . Similarly we find . Hence for all in , so certainly . But this means that is not prime.

This easy theorem is our first real taste of the power that abstract algebra lends to the study of geometry. Let’s see it in action.

Recall that a nonzero principal ideal of the ring is prime iff it is generated by an irreducible polynomial. This is an easy consequence of the fact that is a UFD. Indeed a nonzero principal ideal is prime iff it is generated by a prime element. But in a UFD every prime is irreducible, and every irreducible is prime!

Using the theorem we can say that every irreducible polynomial gives rise to an irreducible affine hypersurface s.t. . Note that we cannot get a converse to this – there’s nothing to say that must be principal in general.

Does this generalise to ideals generated by several irreducible polynomials? We quickly see the answer is no. Indeed take in . These are both clearly irreducible, but is not prime. We can see this in two ways. Algebraically but . Geometrically, recall Lemma 2.5 (3). Also note that by definition . Hence . But is clearly just two distinct points (the intersection of the line with the circle). Hence it is reducible, and by our theorem cannot be prime.

We can also use the theorem to exhibit a more informative example of a reducible variety. Consider . Clearly is not prime for but . Noting that we see that geometrically is the union of the -axis and the parabola , by Lemma 2.5.

Having had such success with prime ideals and irreducible varieties, we might think – what about maximal ideals? Turns out that they have a role to play too. Note that maximal ideals are automatically prime, so any varieties they generate will certainly be irreducible.

Definition 3.9 An ideal of is said to be maximal if whenever either or . Equivalently is maximal iff is a field.

Theorem 3.10 An affine variety in is a point iff is a maximal ideal.

Proof [““] Let be a single point. Then clearly . But a field. Indeed isomorphic to itself, via the isomorphism . Hence maximal.

[““] We’ll see this next time. In fact all we need to show is that are the only maximal ideals.

Theorems 3.8 and 3.10 are a promising start to our search for a dictionary between algebra and geometry. But they are unsatisfying in two ways. Firstly they tell us nothing about the behaviour of reducible affine varieties – a very large class! Secondly it is not obvious how to use 3.8 to construct irreducibly varieties in general. Indeed there is an inherent asymmetry in our knowledge at present, as I shall now demonstrate.

Given an irreducible variety we can construct it’s ideal and be sure it is prime, by Theorem 3.8. Moreover we know by Lemma 2.5 that , a pleasing correspondence. However, given a prime ideal we cannot immediately say that is prime. For in Lemma 2.5 there was nothing to say that , so Theorem 3.8 is useless. We clearly need to find a set of ideals for which holds, and hope that prime ideals are a subset of this.

It turns out that such a condition is satisfied by a class called radical ideals. Next time we shall prove this, and demonstrate that radical ideals correspond exactly to algebraic varieties. This will provide us with the basic dictionary of algebraic geometry, allowing us to proceed to deeper results. The remainder of this post shall be devoted to radical ideals, and the promised proof of an irreducible decomposition.

Definition 3.11 Let be an ideal in a ring . We define the radical of to be the ideal . We say that is a radical ideal if .

(That is a genuine ideal needs proof, but this is merely a trivial check of the axioms).

At first glance this appears to be a somewhat arbitrary definition, though the nomenclature should seem sensible enough. To get a more rounded perspective let’s introduce some other concepts that will become important later.

Definition 3.12 A polynomial function or regular function on an affine variety is a map which is defined by the restriction of a polynomial in to . More explicitly it is a map with for all where some polynomial.

These are eminently reasonable quantities to be interested in. In many ways they are the most obvious functions to define on affine varieties. Regular functions are the analogues of smooth functions in differential geometry, or continuous functions in topology. They are the canonical maps.

It is obvious that a regular function cannot in general uniquely define the polynomial giving rise to it. In fact suppose . Then on so . This simple observation explains the implicit claim in the following definition.

Definition 3.13 Let be an affine variety. The coordinate ring is the ring . In other words the coordinate ring is the ring of all regular functions on .

This definition should also appear logical. Indeed we define the space of continuous functions in topology and the space of smooth functions in differential geometry. The coordinate ring is merely the same notion in algebraic geometry. The name ‘coordinate ring’ arises since clearly is generated by the coordinate functions restricted to . The reason for our notation should now be obvious. Note that the coordinate ring is trivially a finitely generated -algebra.

The coordinate ring might seem a little useless at present. We’ll see in a later post that it has a vital role in allowing us to apply our dictionary of algebra and geometry to subvarieties. To avoid confusion we’ll stick to for the near future. The reason for introducing coordinate rings was to link them to radical ideals. We’ll do this via two final definitions.

Definition 3.14 An element of a ring is called nilpotent if some positive integer s.t. .

Definition 3.15 A ring is reduced if is its only nilpotent element.

Lemma 3.16 is reduced iff is radical.

Proof Let be a nilpotent element of i.e. . Hence so by definition . Conversely let . Then in so i.e. . $/blacksquare$

Putting this all together we immediately see that the coordinate ring is a reduced, finitely generated -algebra. That is, provided we assume that for an affine variety , is radical, which we’ll prove next time. It’s useful to quickly see that these properties characterise coordinate rings of varieties. In fact given any reduced, finitely generated -algebra we can construct a variety with as follows.

Write and define a surjective homomorphism . Let and . By the isomorphism theorem so is radical since reduced. But then by our theorem next time an affine variety, with coordinate ring .

We’ve come a long way in this post, and congratulations if you’ve stayed with me through all of it! Let’s survey the landscape. In the background we have abstract algebra – systems of equations whose solutions we want to study. In the foreground are our geometrical ideas – affine varieties which represent solutions to the equations. These varieties are built out of irreducible blocks, like Lego. We can match up ideals and varieties according to various criteria. We can also study maps from geometrical varieties down to the ground field using the coordinate ring.

Before I go here’s the promised proof that irreducible varieties really are the building blocks we’ve been talking about.

Theorem 3.17 Every affine variety has a unique decomposition as up to ordering, where the are irreducible components and for .

Proof (Existence) An affine variety is either irreducible or with proper subset of . We similarly may decompose and if they are reducible, and so on. We claim that this process stops after finitely many steps. Suppose otherwise, then contains an infinite sequence of subvarieties . By Lemma 2.5 (5) & (7) we have . But a Noetherian ring by Hilbert’s Basis Theorem, and this contradicts the ascending chain condition! To satisfy the condition we simply remove any such that exist in the decomposition we’ve found.

(Uniqueness) Suppose we have another decomposition with for . Then . Since is irreducible we must have for some . In particular . But now by doing the same with the and reversed we find width . But this forces and . But was arbitrary, so we are done.

If you’re interested in calculating some specific examples of ideals and their associated varieties have a read about Groebner Bases. This will probably become a topic for a post at some point, loosely based on the ideas in Hassett’s excellent book. This question is also worth a skim.

I leave you with this enlightening MathOverflow discussion , tackling the irreducibility of polynomials in two variables. Although some of the material is a tad dense, it’s nevertheless interesting, and may be a useful future reference!

Yes and I’ll have another one of those please waiter. One shot Geometry, topped up with Algebra and then a squeeze of Topology. Shaken, not stirred.

Okay, I admit that was both clichéd and contrived. But nonetheless it does accurately sum up the content of this post. We’ll shortly see that studying affine varieties on their own is like having a straight shot of gin – a little unpleasant, somewhat wasteful, and not an experience you’d be keen to repeat.

Part of the problem is the large number of affine varieties out there! We took a look at some last time, but it’s useful to have just a couple more examples. An affine plane curve is the zero set of any polynomial in . These crop up all the time in maths and there’s a lot of them. Go onto Wolfram Alpha and type plot f(x,y) = 0 replacing the term f(x,y) with any polynomial you wish. Here are a few that look nice

There’s a more general notion than an affine plane curve that works in . We say a hypersurface is the zero of a single polynomial in . The cone in that we say last time is a good example of a hypersurface. Finally we say a hyperplane is the zero of a single polynomial of degree in .

Hopefully all that blathering has convinced you that there really are a lot of varieties, and so it looks like it’s going to be hard to say anything general about them. Indeed we could look at each one individually, study it hard and come up with some conclusions. But to do this for every single variety would be madness!

We could also try to group them into different types, then analyse them geometrically. This way is a bit more efficient, and indeed was the method of the Ancients when they learnt about conic sections. But it is predictably difficult to generalise this to higher dimensions. Moreover, most geometrical groupings are just the tip of the iceberg!

What with all this negativity, I imagine that a shot of gin sounds quite appealing now. But bear with me one second, and I’ll turn it into a Long Island Iced Tea! By broadening our horizons a bit with algebraic and topological ideas, we’ll see that all is not lost. In fact there are deep connections that make our (mathematical) life much easier and richer, thank goodness.

First though, I must come good on my promise to tell you about some subset’s of that aren’talgebraic varieties. A simple observation allows us to come up with a huge class of such subsets. Recall that polynomials are continuous functions from to , and therefore their zero sets must be closed in the Euclidean topology. Hence in particular, no open ball in can be thought of as a variety. (If you didn’t understand this, it’s probably time to brush up on your topology).

There are two further ‘obvious’ classes. Firstly graphs of transcendental functions are not algebraic varieties. For example the zero set of the function is not an affine variety. Secondly the closed square is an example of a closed set which is not an affine variety. This is because it clearly contains interior points, while no affine variety in can contain such points. I’m not entirely sure at present why this is, so I’ve asked on math.stackexchange for a clarification!

How does algebra come into the mix then? To see that, we’ll need to recall a definition about a particular type of ring.

Definition 2.1 A Noetherian ring is a ring which satisfies the ascending chain condition on ideals. In other words given any chain s.t. for all .

It’s easy to see that all fields are trivially Noetherian, for the only ideals in are $latex 0$ and itself. Moreover we have the following theorem due to Hilbert, which I won’t prove. You can find the (quite nifty) proof here.

Theorem 2.2 (Hilbert Basis) Let be Noetherian. Then is Noetherian also, and by induction so is for any positive integer .

This means that our polynomial rings will always be Noetherian. In particular, we can write any ideal as for some finite , using the ascending chain condition. Why is this useful? For that we’ll need a lemma.

Lemma 2.3 Let be an affine variety, so some . Let , the ideal generated by . Then .

Proof By definition so . We now need to show the reverse inclusion. For any there exist polynomials in and in s.t. . Hence if then so .

Let’s put all these ideas together. After a bit of thought, we see that every affine variety can be written as the zero set of a finite number of polynomials . If you don’t get this straight away look back carefully at the theorem and the lemma. Can you see how to marry their conclusions to get this fact?

This is an important and already somewhat surprising result. If you give me any subset of obtained from the solutions (possibly infinite) number of polynomial equations, I can always find a finite number of equations whose solutions give your geometrical shape! (At least in theory I can – doing so in practice is not always easy).

You can already see that a tiny bit of algebra has sweetened the cocktail! We’ve been able to deduce a fact about every affine variety with relative ease. Let’s pursue this link with algebra and see where it takes us.

Definition 2.4 For any subset we say the ideal of is the set .

In other words the ideal of is all the polynomials which vanish on the set . A trivial example is of course . Try to think of some other obvious examples before we move on.

Let’s recap. We’ve now defined two maps and . Intuitively these maps are somehow ‘opposite’ to each other. We’d like to be able to formalise that mathematically. More specifically we want to find certain classes of affine varieties and ideals where and are mutually inverse bijections.

Why did I say certain classes? Well, clearly it’s not the case that and are bijections on their domains of definition. Indeed , but so isn’t injective. Furthermore working in we see that so is not injective. Finally for so is not surjective.

It’ll turn out in the next post that a special type of ideal called a radical ideal will play an important role. To help motivate its definition, think of some more examples where fails to be injective. Can you spot a pattern? We’ll return to this next time.

Now that we’ve got our maps and it’s instructive to examine their properties. This will give us a feeling for the basic manipulations of algebraic geometry. No need to read it very thoroughly, just skim it to pick up some of the ideas.

Lemma 2.5 The maps and satisfy the following, where ideals and subsets of :
(1)
(2)
(3)
(4)
(5)
(6)
(7) with equality iff is an affine variety

Proof We prove each in turn.
(1) Trivial.
(2) We first prove ““. Let . Wlog assume . Then . So certainly , which is what we needed to prove. Now we show ““. Let . Then and . So there exists s.t. . Hence . But s0 .
(3) “” is trivial. For “” note that , and then it’s trivial.
(4) Trivial.
(5) Trivial.
(6) If then by definition, so .
(7) The relation follows from definitions exactly as (6) did. For the “if” statement, suppose , some ideal . Then by (5) so by (4) . Conversely, suppose . Then is the zero set of so an affine variety by definition.

That was rather a lot of tedious set up! If you’re starting to get weary with this formalism, I can’t blame you. You may be losing sight of the purpose of all of this. What are these maps and and why do we care how they behave? A fair question indeed.

The answer is simple. Our bijections will give us a dictionary between algebra and geometry. With minimal effort we can translate problems into an easier language. In particular, we’ll be allowed to use a generous dose of algebra to sweeten the geometric cocktail! You’ll have to wait until next time to see that in all its glory.

Finally, how does topology fit into all of this? Well, Lemma 2.5 (1)-(3) should give you an inkling. Indeed it instantly shows that the following definition makes sense.

Definition 2.6 We define the Zariski topology on by taking as closed sets all the affine varieties.

In some sense this is the natural topology on when we are concerned with solving equations. Letting we can make some comparisons with the usual Euclidean topology.

First note that since every affine variety is closed in the Euclidean topology, every Zariski closed set is Euclidean closed. However we saw in the last post that not all Euclidean closed sets are affine varieties. In fact there are many more Euclidean closed sets than Zariski ones. We say that the Euclidean topology is finer than the Zariski topology. Indeed the Euclidean topology has open balls of arbitrarily small radius. The general Zariski open set is somehow very large, since it’s the complement of a line or surface in .

Next time we’ll prove that for algebraically closed every Zariski open set is dense in the Zariski topology, and hence (if ) in the Euclidean topology. In particular, no nonempty Zariski open set is bounded in the Euclidean topology. Hence we immediately see that the intersection of two nonempty Zariski open sets of ^n is never empty. This important observation tells us the the Zariski topology is notHausdorff. We really are working with a very strange topological space!

And how is this useful? You know what I am going to say. It gives us yet another perspective on the world of affine varieties! Rather than just viewing them as geometrical objects in abstract we can imagine them as a fundamental world structure. We’ll now be able to use the tools of topology to help us learn things about geometry. And there’s the slice of lemon to garnish the perfect cocktail.

I leave you with this enlightening question I recently stumbled upon. Both the question, and the proposed solutions struck me as extremely elegant.

Time to introduce some real mathematics! Today we’ll be talking about algebraic varieties. Gosh, that already sounds pretty heavy going. Part of the problem with starting algebraic geometry is the none of the nomenclature makes any intuitive sense. So it’s probably worth going on a bit of a historical digression to find out where this term originated.

Back in the 19th century, a good deal of algebraic geometry was done by French mathematicians. So it’s not surprising that much of the terminology of basic algebraic geometry has been borrowed from French. The word variety is one example. In 19th century French, a variété was an umbrella term for a geometrical object in space. A typical example of a 19th century variété would be a manifold, that is a space that looks locally like everywhere.

As time passed, the word variété caught on in English, despite the fact it seemed linguistically arcane. Mathematicians rarely worry about such things, it seems. As maths became increasingly formalised and rigorous, new terms like manifold and surface were introduced to describe particular types of varieties. By a combination of French stubbornness and historical accident, the word variety eventually came to refer to an abstract class of geometrical ‘things’.

(If you want more information, read this wonderful account of the history of Algebraic Geometry).

Hopefully the concept of algebraic presents fewer difficulties. As I mentioned in an earlier post, algebra is essentially the study of solutions to (mostly polynomial) equations. So what’s an algebraic variety? You got it – it’s a geometrical object which can be represented as a solution of (one or many) polynomial equations.

I’d properly better formalise all that as a definition. But first we need to know what kind of space we are working in. In other words, where do we allow our algebraic varieties to exist? The naive answer is in -dimensional Euclidean space. This is indeed a good suggestion, and yields many informative examples, but there is too much loss of generality. Instead we’ll work in -dimensional affine space which I’ll define shortly. Keep the idea of -dimensional Euclidean space in mind as an intuition, though!

Definition 1.1 Let be a field. We say affine space of dimension over is the set .

You might think this is a bit of an odd notation. After all it takes more time to write than and they are the same as sets by definition. However there is a subtlety. Mathematicians often think of as being endowed with a natural vector space structure, with an origin and addition operation. Affine space is to be regarded merely as a geometrical blank canvas, with no associated operations or distinguished points. In fact we’ll see later that the right way to think about is as a topological space.

Since this post has an historical bent, I’ll digress a little to explain why we use the word affine. The word has its roots in Latin – affinis, meaning ‘related’. Mathematical usage seems to have been introduced by Euler to describe a type of geometry that studies how geometric objects are ‘related’ by slanting and scaling. Absolute notions of length and angle cease to make sense in this setting. Rather affine geometry is concerned more with the concepts of parallelism and ratios of lengths.

This might all seem a bit abstract, so let me put it another way. Affine geometry is the study of shapes which remain unchanged when they are transformed in such a way as to preserve straight lines. These so called affine transformations crop up all the time – translation, expansion, rotation are all examples we meet in everyday life. Affine geometry tries to make sense of all these in one geometrical space, affine space.

Definition 1.2Let be a subset of the polynomial ring. We define the zero locus of to be the set .

Sorry if that definition was a bit out of the blue. You may have to get used to me moving fast as this blog evolves. Remember that the polynomial ring is just the set of all polynomials in the variables endowed with the obvious addition operation allowing you to add two polynomials. In plain English this definition is saying, ‘the zero locus of a set of polynomials, is all the points that make all the polynomials zero’. Sensible, huh?

Let’s do some examples. If we fix and work in we have is a circle. Can you see what is? (Hint: the answer is in an earlier post)! Now if we work in we can get some familiar surfaces. is a cone. is a weird shape called a twisted cubic, pictured below.

Have some fun trying to think up some more wacky and wonderful shapes that can be represented as the zero locus of a subset of . (Note: we’ll sometimes use the terminology as I have done here). Do leave me a comment if you come up with something fun!

Finally we’re ready to say what an algebraic variety is. Here we go.

Definition 1.3 A subset is called an affine algebraic variety if there exists some subset of the polynomial ring such that .

Read that a couple of times and make sure you understand it. This really is the bedrock on which the subject stands. In plain English this merely says that ‘an affine algebraic variety is a geometrical shape which can be represented as the zero locus of some polynomials’. That’s exactly what we said it should mean above. (Note: I’ll often call affine algebraic varieties just affine varieties for short).

Right, that’s quite enough for one night. Next time I’ll talk about what Hilbert had to say about affine varieties. We’ll also start to see a surprisingly deep connection between algebra and geometry. Oh and if someone reminds me I’ll throw in an amusing video, like this marvellous CassetteBoy offering.

One more thing to do before you go – think about what kinds of shapes aren’t affine varieties. Answers in the comments please! Looking for such examples is something mathematicians like to do. It’ll hopefully give you a better understanding of what the concepts really mean! I’ll touch on this properly next time.

Apologies that this post is a little late – I have been struggling with some technical issues! I hope now they are sorted, thanks to the kindness of the IT Department at New College, Oxford.

and a nice cup of tea. I always find that helps. Before we get down to business, you might want to put this delightful recording on. It’s always nice to have a bit of background music, and Strauss just seems to fit with Algebra somehow.

A broad definition of Algebra could be the study of equations and their solutions. This is perhaps the type of algebra we’re all familiar with from school. Here’s a typical problem

Find given that

That was easy, of course. Let’s try another one

Find all such that

Perhaps you had to think for a moment before realising that this just defines a parabola in 2D space, pictured below.

These example illustrate that the solutions to equations can come in the form of points, or curves, and it’s not hard to see that solutions to equations in sufficiently many variables can define surfaces of any dimension you like. For example the equation defines a plane in 3D space.

So we can easily see that Algebra gives rise to geometrical structures of the type we discussed in the last post. It should now seem natural to study geometrical structures from an algebraic point of view. Voila – we have the motivation for Algebraic Geometry.

There’s nothing to restrict us to studying the solutions (often referred to as zeroes) of a single equation. In fact many interesting and useful geometric constructions arise as the simultaneous zeroes of several equations. Can you see two equations in (some or all of) the variables whose simultaneous solutions give rise to the -axis in 3D [2]?

The technical terminology for the collection of simultaneous zeroes of several equations is an algebraic set. It is the most fundamental object of study which we will focus on.

Here we reach a slight technical impasse. For what follows I’ll assume a familiarity with elementary abstract algebra as outlined on the Background page. This may be viewed as a technical toolkit for our forthcoming studies. I’ll also assume some very basic knowledge about Topology, though not much more than can be gleaned by a thorough reading of the Wikipedia page. If you’ve never come across abstract algebra before, now is the time to do some serious thinking! I can’t promise it’ll be easy, and it might take a couple of days to get your head around the concepts, but I promise you it’s worth it. I’ll be happy to answer any questions commented on the Background page, and will flesh out the currently sparse details in the near future.

Good luck!

[1] We only every consider polynomial equations, which are those of the form where is a finite sum of nonnegative integer powers of products of the variables . Thus , the circle, is admissible for study but is not. It turns out that not much is lost by restricting our study to polynomials only. In some sense any mathematically interesting curve can be approximated arbitrarily closely by the set of solutions to polynomial equations. (This entirely depends on your definition of mathematically interesting though)!

[2] The equations are of course and . Geometrically this is true since the -axis is the intersection of the two planes defined by and .

Okay it’s a bit of a mouthful. Let’s break it down a bit. You probably remember geometry from school. Drawing triangles and calculating angles. Maybe even a few circle theorems. Pretty arcane stuff, you probably agree. Turns out that this is just one tiny area of what mathematicians call Geometry.

Roughly speaking Geometry is the study of any kinds of curves, shapes and surfaces you can imagine. We naturally think of curves as “1-dimensional objects” you can draw on a “2-dimensional” piece of paper. Similarly we think of surfaces as “2 dimensional objects” that exist in “3-dimensional space”. A piece of paper is an example of a surface, as it the surface of a beach ball. We can also think of “3-dimensional objects” like a solid snooker ball. In general geometry answers questions about what properties these things have.

Now you may be thinking that this is all a bit pointless. After all we know quite a lot about how a beach ball behaves. There are two caveats however. Firstly the surface of a beach ball is a very symmetrical object. We want to be able to make conclusions about vastly asymmetrical surfaces, and possibly ones it’s hard to imagine. These kind of general observations are useful because then if someone asks us about a specific case, a beach ball with a ring donut stuck onto it for example, we can tell them it’s geometrical features with no extra work.

The second pertinent observation is that sometimes we want to know about geometry in more than “3-dimensions”. Hang about, that’s completely pointless, I hear you say. Fair point, but you are forgetting we live in a 4-dimensional universe – 3 space and 1 time dimension. And thanks to Einstein’s General Theory of Relativity we know that gravity bends space. So knowing about how geometry works in 4D is vital for sending men to the moon, or getting accurate GPS signals from satellites.

Now you might be starting to see that Geometry is quite broad, quite useful but also quite hard. After all there doesn’t seem to be much a triangle has in common with the surface of the Earth! Nevertheless we’ll see in the next post that using Algebra we can start to pin down some classes of curves and surfaces that do share some surprisingly strong properties.

As a challenge before the next post, make a Mobius Strip, pictured above, and count how many sides it has. Okay that’s easy, if a little odd if it’s the first time you’ve seen it. You can see that even an easily constructible surface can have some surprises. Try and imagine some other odd surfaces; if you think of anything good then comment it! One such is the Klein Bottle. Don’t worry about the technical terminology on the Wikipedia page, just have a look at the pictures. It’s a 2-dimensional surface that can only be “drawn” in 4-dimensions. Looks like we’ll have our work cut out!